Truncated Conical Indenter

Indentation of an elastic flat by a truncated cone

The truncated conical intender is characterized by the radius of the blunt end b, the inclination angle \theta, the effective contact radius a and the deformation depth d (rigid body indentation). This solution is correct without restrictions only for rigid indenters (see [1]).
The contact radius a can be deduced numerically using the relation that connects the geometrical parameters to each other:

(1)   \begin{equation*} d(a) = a \tan(\theta) \arccos\left(\frac{b}{a}\right) = \phi_0 a \tan(\theta) \end{equation*}

such that the angle \phi_0 is introduced as:

(2)   \begin{equation*} \phi_0 := \arccos\left(\frac{b}{a}\right) \end{equation*}

Other parameters, such as load force F_N, displacement \omega(r;a) and the stress profile \sigma_{zz}(r;a) are given by the following equations:

(3)   \begin{align*} F_N(a) &= E^* \tan(\theta) a^2 \left[ \arccos\left(\frac{b}{a}\right) + \frac{b}{a} \sqrt{1 - \frac{b^2}{a^2} } \right]\\ &= E^* \tan(\theta) a^2 \left( \phi_0 + \cos \phi_0 \sin \phi_0 \right) \end{align*}

(4)   \begin{align*} \sigma_{zz}(r;a) &= \nonumber \\ &= - \frac{E^* \tan(\theta)}{\pi} \begin{cases} \int_{b}^{a} \left[ \frac{b}{\sqrt{x^2 - b^2}} + \arccos\left(\frac{b}{x}\right)\right] \frac{dx}{\sqrt{x^2 - r^2}} & r \leq b, \\ \int_{r}^{a} \left[ \frac{b}{\sqrt{x^2 - b^2}} + \arccos\left(\frac{b}{x}\right)\right] \frac{dx}{\sqrt{x^2 - r^2}} & b < r \leq a. \end{cases}  \end{align*}

(5)   \begin{align*} \omega(r;a) &= \frac{2\tan(\theta)}{\pi} \left\{ \phi_0 a \arcsin\left(\frac{a}{r}\right) - \int_b^a x \arccos\left( \frac{b}{x}\right) \frac{dx}{\sqrt{r^2 - x^2}} \right\, r>a} \end{align*}


Poisson’s ratio of the substrate \nu dimensionless,
Young’s modulus of elasticity E of the substrate, [Pa],
Equivalent elastic constant  E^* = \left( \frac{1-\nu^2}{E} \right)^{-1}, [Pa],
Normal load F_N, [N]


[1] Valentin L. Popov, Hanbook of Contact Mechnics, Exact Solutions of Axisymmetric Contact Problems, pg. 13, pg.29